Unravelling the complex correlation between chemical shifts of 13 C α, 13 C β, 13 C′, 1 H α, 15 N, 1 H N atoms in amino acids of proteins from NMR experiment and local structural environments of amino acids facilitates the assignment of secondary structures of proteins. This is an important impetus for both determining the three-dimensional structure and understanding the biological function of proteins. The previous empirical correlation scores which relate chemical shifts of 13 C α, 13 C β, 13 C′, 1 H α, 15 N, 1 H N atoms to secondary structures resulted in progresses toward assigning secondary structures of proteins. However, the physical-mathematical framework for these was elusive partly due to both the limited and orthogonal exploration of higher-dimensional chemical shifts of hetero-nucleus and the lack of physical-mathematical understanding underlying those correlation scores. Here we present a simple multi-dimensional hetero-nuclear chemical shift score function (MDHN-CSSF) which captures systematically the salient feature of such complex correlations without any references to a random coil state of proteins. We uncover the symmetry-breaking vector and its reliability order not only for distinguishing different secondary structures of proteins but also for capturing the delicate sensitivity interplayed among chemical shifts of 13 C α, 13 C β, 13 C′, 1 H α, 15 N, 1 H N atoms simultaneously, which then provides a straightforward framework toward assigning secondary structures of proteins. MDHN-CSSF could correctly assign secondary structures of training (validating) proteins with the favourable (comparable) Q3 scores in comparison with those from the previous correlation scores. MDHN-CSSF provides a simple and robust strategy for the systematic assignment of secondary structures of proteins and would facilitate the de novo determination of three-dimensional structures of proteins.
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